Search efficiency of discrete fractional Brownian motion in a random distribution of targets

Abstract

Efficiency of search for randomly distributed targets is a prominent problem in many branches of the sciences. For the stochastic process of Lévy walks, a specific range of optimal efficiencies was suggested under variation of search intrinsic and extrinsic environmental parameters. In this paper, we study fractional Brownian motion as a search process, which under parameter variation generates all three basic types of diffusion, from sub- to normal to superdiffusion. In contrast to Lévy walks, fractional Brownian motion defines a Gaussian stochastic process with power-law memory yielding antipersistent, respectively persistent motion. Computer simulations of search by time-discrete fractional Brownian motion in a uniformly random distribution of targets show that maximising search efficiencies sensitively depends on the definition of efficiency, the variation of both intrinsic and extrinsic parameters, the perception of targets, the type of targets, whether to detect only one or many of them, and the choice of boundary conditions. In our simulations, we find that different search scenarios favor different modes of motion for optimising search success, defying a universality across all search situations. Some of our numerical results are explained by a simple analytical model. Having demonstrated that search by fractional Brownian motion is a truly complex process, we propose an overarching conceptual framework based on classifying different search scenarios. This approach incorporates search optimization by Lévy walks as a special case.